Tool to make probabilities on picking objects. Calculation of probabilities of drawing objects (balls, beads, cards, etc.) in a box (bag, drawer, deck, etc.) with and without replacement is a common exercise in probability.

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Probabilities for a Draw with Replacement

Tool to make probabilities on picking objects. Calculation of probabilities of drawing objects (balls, beads, cards, etc.) in a box (bag, drawer, deck, etc.) with and without replacement is a common exercise in probability.

Answers to Questions

How to compute a probability of picking without replacement?

For a set of \( N \) objects among which \( m \) are different (distinguishable). The probability of drawing a total of \( n \) objects and that among these \( n \) objects there are \( k \) objects that are part of the \( m \) different ones, is given by a hypergeometric distribution: $$ p(X=k)=\frac{C_{m}^kC_{N-m}^{n-k}}{C_N^n} = \frac{ \binom{m}{k} \binom{N-m}{n-k} }{ \binom{N}{n} } $$